Show commands:
SageMath
E = EllipticCurve("gc1")
E.isogeny_class()
Elliptic curves in class 187200gc
sage: E.isogeny_class().curves
| LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
|---|---|---|---|---|---|---|---|---|
| 187200.pn4 | 187200gc1 | \([0, 0, 0, -2065800, -963493000]\) | \(83587439220736/13990184325\) | \(163181509966800000000\) | \([2]\) | \(4718592\) | \(2.5996\) | \(\Gamma_0(N)\)-optimal |
| 187200.pn2 | 187200gc2 | \([0, 0, 0, -31590300, -68338402000]\) | \(18681746265374416/693005625\) | \(129331481760000000000\) | \([2, 2]\) | \(9437184\) | \(2.9461\) | |
| 187200.pn3 | 187200gc3 | \([0, 0, 0, -30132300, -74931478000]\) | \(-4053153720264484/903687890625\) | \(-674599395600000000000000\) | \([2]\) | \(18874368\) | \(3.2927\) | |
| 187200.pn1 | 187200gc4 | \([0, 0, 0, -505440300, -4373739502000]\) | \(19129597231400697604/26325\) | \(19651507200000000\) | \([2]\) | \(18874368\) | \(3.2927\) |
Rank
sage: E.rank()
The elliptic curves in class 187200gc have rank \(0\).
Complex multiplication
The elliptic curves in class 187200gc do not have complex multiplication.Modular form 187200.2.a.gc
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rrrr} 1 & 2 & 4 & 4 \\ 2 & 1 & 2 & 2 \\ 4 & 2 & 1 & 4 \\ 4 & 2 & 4 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.
